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Unformatted text preview: garcia (jjg2564) – HW 10 – gualdani – (56410) 1 This printout should have 23 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points Let f be the function defined by f ( x ) = 3 + 2 x 1 / 3 . Consider the following properties: A. derivative exists for all x negationslash = 0 ; B. concave up on (0 , ∞ ) ; C. has horizontal tangent at x = 0 . Which does f have? 1. B only 2. A only correct 3. None of them 4. B and C only 5. A and B only 6. A and C only 7. C only 8. All of them Explanation: The graph of f is 2 4 − 2 − 4 2 4 6 On the other hand, after differentiation, f ′ ( x ) = 2 3 x 2 / 3 , f ′′ ( x ) = − 4 9 x 5 / 3 . Consequently, A. has: ( f ′ ( x ) = (2 / 3) x − 2 / 3 , x negationslash = 0); B. not have: ( f ′′ ( x ) < , x > 0); C. not have: (see graph). 002 10.0 points If f is increasing and its graph is concave down on (0 , 1), which of the following could be the graph of the derivative , f ′ , of f ? 1. 1 f ′ ( x ) 2. f ′ ( x ) 1 3. 1 f ′ ( x ) garcia (jjg2564) – HW 10 – gualdani – (56410) 2 4. f ′ ( x ) 1 cor rect Explanation: The function f increases when f ′ > 0 on (0 , 1), and its graph is concave down when f ′′ < 0. Thus on (0 , 1) the graph of f ′ lies above the xaxis and is decreasing. Of the four graphs, only f ′ ( x ) 1 has these properties. 003 10.0 points When Sue uses first and second derivatives to analyze a particular continuous function y = f ( x ) she obtains the chart y y ′ y ′′ x < − 3 + − x = − 3 4 − 3 < x < − − x = 0 1 − 1 < x < 2 − + x = 2 − 1 DNE x > 2 + + Which of the following can she conclude from her chart? A. f has a point of inflection at x = 2. B. f is concave down on ( −∞ , 0) . C. f is concave up on (0 , 2) . 1. none of them 2. A only 3. all of them 4. C only 5. B only 6. B and C only correct 7. A and C only 8. A and B only Explanation: The graph of f must look like 2 − 2 − 4 2 4 Consequently, A. False. B. True. C. True. 004 10.0 points The figure below shows the graphs of three functions: garcia (jjg2564) – HW 10 – gualdani – (56410) 3 One is the graph of a function f , one is its derivative f ′ , and one is its second derivative f ′′ . Identify which graph goes with which function. 1. f : f ′ : f ′′ : 2. f : f ′ : f ′′ : 3. f : f ′ : f ′′ : 4. f : f ′ : f ′′ : 5. f : f ′ : f ′′ : correct 6. f : f ′ : f ′′ : Explanation: Calculus tells us that f (i) has horizontal tangent at ( x , f ( x )) when f ′ crosses the xaxis, (ii) is increasing when f ′ > 0, and (iii) is decreasing when f ′ < 0, (iv) has a local max at x when f ′ ( x ) = 0 and f ′′ ( x ) < 0, (v) has a local min at x when f ′ ( x ) = 0 and f ′′ ( x ) > 0, (vi) is concave up when f ′′ > 0, (v) and concave down when f ′′ < 0....
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 Fall '09
 Gualdani
 Derivative, lim

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